Hamilton cycle in small d-diregular graphs

An directed graph is -diregular if every vertex has indegree and outdegree at least .

Conjecture For , every -diregular oriented graph on at most vertices has a Hamilton cycle.

The disjoint union of two regular tournaments on vertices shows that this would be best possible. For -diregular oriented graphs with an arbitrary order of vertices, Jackson conjectured the existence of a long cycle.

Kühn and Osthus [KO] conjectured that it may actually be possible to increase the size of the graph even further if we assume that the graph is strongly 2-connected.

Problem Is it true that for each , every -regular strongly -connected oriented graph on at most vertices has a Hamilton cycle?